Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-3x-6y &= 9 \\ -2x-7y &= 7\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $3$ $\begin{align*}6x+12y &= -18\\ -6x-21y &= 21\end{align*}$ Add the top and bottom equations. $-9y = 3$ Divide both sides by $-9$ and reduce as necessary. $y = -\dfrac{1}{3}$ Substitute $-\dfrac{1}{3}$ for $y$ in the top equation. $-3x-6( -\dfrac{1}{3}) = 9$ $-3x+2 = 9$ $-3x = 7$ $x = -\dfrac{7}{3}$ The solution is $\enspace x = -\dfrac{7}{3}, \enspace y = -\dfrac{1}{3}$.